NZ Level 7 (NZC) Level 2 (NCEA)
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Area of Circles and Sectors
Lesson

We already know that area is the space inside a 2D shape.  We can find the area of a circle, but we will need a special rule.  

The following investigation will demonstrate what happens when we unravel segments of a circle.  

Interesting isn't it that when we realign the segments we end up with a parallelogram shape.  Which is great, because it means we know how to find the area based on our knowledge that the area of a parallelogram has formula $A=bh$A=bh.  In a circle, the base is half the circumference and the height is the radius.  

 

Area of a Circle

$\text{Area of a circle}=\pi r^2$Area of a circle=πr2

What if we don't have an entire circle?  

Well, half a circle would have half the area or half the circumference. One quarter of a circle would have a quarter of the area, or a quarter of the circumference.  In fact all we need to know is what fraction the sector is of a whole circle.  For this all we need to know is the angle of the sector.

Looking at the quarter circle, the angle of the sector is $90$90°. The fraction of the circle is $\frac{90}{360}=\frac{1}{4}$90360=14

More generally, If the angle of the sector is $\theta$θ, then the fraction of the circle is represented by

$fraction=\frac{\theta}{360}$fraction=θ360 (due to there being $360$360° in a circle).

 

Example

Question: Find the area of a sector with central angle of $126$126° and radius of $7$7cm. Evaluate to $2$2 decimal places.  

Think: What fraction is this sector of a whole circle?  What is the rule for area?

Do:  This sector is $\frac{126}{360}=0.35$126360=0.35 of a circle.

Area of a circle is $A=\pi r^2$A=πr2, so the area of the sector is

$0.35\times\pi r^2$0.35×πr2 $=$= $0.35\pi\times7^2$0.35π×72
  $=$= $17.15\pi$17.15π
  $=$= $53.88$53.88 cm2  (rounded to 2 decimal places)

 

More Worked Examples

Question 1

If the radius of the circle is $5$5 cm, find its area.

Give your answer as an exact value.

Question 2

Find the area of the shaded region in the following figure, correct to 1 decimal place.

Question 3

Find the area of the shaded region in the following figure, correct to 1 decimal place.

Question 4

Consider the sector below.

  1. Calculate the perimeter. Give your answer correct to $2$2 decimal places.

  2. Calculate the area. Give your answer correct to $2$2 decimal places.

Question 5

Consider the sector below.

  1. Calculate the perimeter. Round your answer to two decimal places.

  2. Calculate the area. Round your answer to two decimal places.

QUESTION 6

A goat is tethered to a corner of a fenced field (shown). The rope is $9$9 m long. What area of the field can the goat graze over?

  1. Give your answer correct to 2 decimal places.

Outcomes

M7-4

Apply trigonometric relationships, including the sine and cosine rules, in two and three dimensions

91259

Apply trigonometric relationships in solving problems

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